Lecture1-SSP-2007

Welcome to Phys 446: Solid State Physics / Optical PropertiesFall 2007

Instructor:Andrei Sirenko Associate Professor at the Dept. of Physics, NJIT http://web.njit.edu/~sirenko 476 Tiernan Office hours: Class Schedule: Friday After the classes or by appointment 973-596-5342

2:30pm - 5:25pm

| TIER-108

Lecture 1

Andrei Sirenko, NJIT

1

Lecture 1


2

Course Elements:Plan for today: Introduction to the Course Crystals Lab tourTextbooks: M. A. Omar, Elementary Solid State Physics, Addison-Wesley, 1993. Charles Kittel, Introduction to Solid State Physics, 8th Edition, Wiley, 2004. Supplemental texts: H. Ibach, H. Lth, Solid-State Physics. An Introduction to Principles of Materials Science, Springer, 2003. J. S. Blakemore, "Solid State Physics, Third Edition, Cambridge University Press, 1985 P. Yu and M. Cardona, Fundamentals of semiconductors N. W. Ashcroft and N. D. Mermin, Solid State Physics Lecture Slides Demonstrations in the Experimental Lab at NJIT Raman Scattering in Diamond High-Resolution X-ray Diffraction in Silicon wafer Transmission in InP-based multilayer device structure Micro-beam Photoluminescence in InGaAsP-based waveguide device structure Grade Components: Homework: 10 % Research project: 10 % Two in-class exams: 15% each; Final exam: 50%

Lecture 1


3

Lecture 1


4

Course Goals:This course integrates theory of Solid State Physics with experimental demonstrations in the Research Physics Lab. The course will provide a valuable theoretical introduction and an overview of the fundamental applications of the physics of solids. This course includes theoretical description of crystal and electronic structure, lattice dynamics, and optical properties of different materials (metals, semiconductors, dielectrics, magnetic materials and superconductors), based on the classical and quantum physics principles. Several advanced experiments of X-ray diffraction, Raman Scattering, Photoluminescence, etc., will be carrier out in the Research Physics Lab followed by their theoretical discussion.

To help you with your research projects and future jobs

Lecture 1


5

Lecture 1


6

Outline of the course: I. Crystal structure, symmetry and types of chemical bonds. (Chapter 1) The crystal lattice Point symmetry The 32 crystal classes Types of bonding (covalent, ionic, metallic bonding; hydrogen and van der Waals). II. Diffraction from periodic structures (Chapter 2) Reciprocal lattice; Brillouin zones Laue condition and Bragg law Structure factor; defects Methods of structure analysis HRXRD. Experimental demonstration in the Physics Lab using Bruker D8 Discover XRD III. Lattice vibrations and thermal properties (Chapter 3) Elastic properties of crystals; elastic waves Models of lattice vibrations Phonons Theories of phonon specific heat; thermal conduction. Anharmonicity; thermal expansion Raman Scattering by phonons. Experimental demonstration in the Physics Lab using Ar-laser/SPEX 500M, CCD based Raman Scattering setup IV. Electrons in metals (Chapters 45) Free electron theory of metals Fermi Statistics Band theory of solids V. Semiconductors (Chapters 67) Band structure. Electron statistics; carrier concentration and transport; conductivity; mobility Impurities and defects Magnetic field effects: cyclotron resonance and Hall effect Optical properties; absorption, photoconductivity and luminescence Basic semiconductor devices Photoluminescence. Experimental demonstration in the Physics Lab using Nd:YAG laser/SPEX based Photoluminescence setup VI. Dielectric properties of solids (Chapters 8) Dielectric constant and polarizability (susceptibility) Dipolar polarizability, ionic and electronic polarizability Piezoelectricity; pyro- and ferroelectricity Light propagation in solids VII. Magnetism (Chapters 9) Magnetic susceptibility Classification of materials; diamagnetism, paramagnetism Ferromagnetism and antiferromagnetism Magnetic resonance Multiferroic Materials VIII. Superconductivity (Chapter 10)

Lecture 1


7

Lecture 1


8

Lectures:Presentation of the concepts and techniques of SSP. Demonstrations. Lecture quiz (periodically) Lectures are not a substitute for reading the textbook and independent literature search Read ahead; youll get more from lecture. Slides will be posted on the course web. Use these as a study guide/note taking aid

Scope and Requirements

Required for Phys-774

PHYS 432 (E&M-I ) Familiarity with basic principles of quantum mechanics (Schrdinger equation, wave function, energy levels, spin) Knowledge of basics of statistical physics (classical statistics, Bose-Einstein and Fermi-Dirac statistics)

Lecture 1


9

Lecture 1


10

Exams: Homework Assignments (10 in total) will be due weekly, usually on Fridays. Assignments are due at the beginning of class. Homework problems, lectures, and text readings will form the basis of the exam problems.

There will be two in-class exams and a final exam. Allowed to use lecture notes and formula sheets. Not allowed to discuss the problems with other students. Academic honesty Students are encouraged to: discuss the lectures and textbook material work together on homework problems study together for exams But not allowed to: present other peoples work as your own (including copying another student's homework as well as using of problem solutions found in the Internet or elsewhere) discuss problems or questions during examsLecture 1 Andrei Sirenko, NJIT 12

ProjectStudents will perform a research project on a selected topic of contemporary solid state physics (of their choice) by reviewing scientific journal articles focusing on the effect chosen. A formal report (~7 pages) will be required.

Lecture 1


11

Today: crystal structures Lecture 1-1What do you know about crystals ? They make nice chandeliers !?.. Superman Returns

Lecture 1


13

Lecture 1


14

Today: crystal structures (Omar Ch. 1.1.-1.6.)Crystal: atoms are arranged so that their positions are periodic in all three dimensions Atoms are bound to one another well defined equilibrium separations; many identical atoms minimum energy requires every identical atom to be in identical environment 3D periodicity Ideal crystal: perfect periodicity Real crystals are never perfect: surface impurities and defects thermal motion of atoms (lattice vibrations)Lecture 1 Andrei Sirenko, NJIT 15 Lecture 1 Andrei Sirenko, NJIT 16

rn = n1 a1 + n2 a2 + n3 a3

Lecture 1


17

Lecture 1


18

Basic definitionsThe periodic array of points is called crystal lattice. For every lattice point there is a group of atoms (or single atom) called basis of the lattice Don't confuse with a1, a2, a3 - basis vectors parallelogram formed by the basis vectors unit cell if a unit cell contains only one lattice point, it is called a primitive cell (minimum volume) Bravais lattices all lattice points are equivalent 2D case 5 Bravais lattices in the next slideLecture 1 Andrei Sirenko, NJIT 19 Lecture 1 Andrei Sirenko, NJIT 20

a3 a2 a1

7 crystal systems and 14 Bravais lattices in 3D triclinic abc monoclinic abc P C a b c, = = = 90 P C I Forthorhombic

a = b c, = = = 90 P I

tetragonal

rhombohedral (trigonal)

cubic

a = b = c, = = 90

= = = 90P I F

a = b = c,

a = b c, = = 90 = 120 simple cubic bcc fcc22

hexagonal

Lecture 1


21

Lecture 1


simple cubic

Lecture 1


23

Lecture 1


24

Lecture 1


25

Lecture 1


26

Lecture 1


27

Lecture 1


28

Lecture 1


29

Lecture 1


30

Examples rock salt perovskite (e.g. BaTiO3) diamond (Si, Ge) zincblende (GaAs ...)

Elements of symmetryIn addition to periodicity (translation) each lattice can have other symmetry properties: Inversion center I Reflection (mirror) plane

Rotation axes. Only 2-, 3-, 4-, and 6-fold rotations are compatiblewith translation invariance

Rotation-inversion axesWhat are the Bravais lattices for these structures? How many atoms are in the basis, and what are their positions? fccLecture 1

cubicAndrei Sirenko, NJIT

fcc31

Point symmetry Every crystal lattice may be described by a particular combination of symmetry operations determined by symmetry of the basis and the symmetry of the Bravais lattice. There are 32 crystal classes (point groups) Combining with translational symmetries, one obtains 230 space groupsLecture 1 Andrei Sirenko, NJIT 32

Elements of symmetry

Symmetry group example: C3v

Inversion center I Reflection (mirror) plane Rotation axes. Only 2-, 3-, 4-, and 6-fold rotations are compatiblewith translation invariance Rotation-inversion axes

Inversion center I Reflection planes: 3 Rotation axes. Only 3- fold rotation

Lecture 1


33

Lecture 1


34

Symmetry group example: O (cube)

Symmetry group example: O (cube)

Inversion center I Reflection planes Rotation axes

total: 24 elements (48 with i: Oh )

Inversion center I Reflection planes Rotation axes

total: 12 elements

Lecture 1


35

Lecture 1


36

Notation for crystallographic directions and planes: Miller indices (see demonstration)Crystal directions:

Summary A crystal lattice is a periodic array of the points related by translation operation:

rn = n1 a1 + n2 a2 + n3 a3

[n1 n2 n3] n1 n2 n3

rn = n1 a1 + n2 a2 + n3 a3

If there are several equivalent directions: Crystal planes:

To form a crystal, we attach to every lattice point an identical group of atoms called basis 7 crystal systems and 14 Bravais lattices 32 crystal classes (point symmetry groups) Crystallographic directions and Miller indices; dhkl

If a plane intercepts the axes at x, y, z and basis vectors are a, b, c, then one can obtain three numbers

a b c , , x y z

Continue with interatomic forces and types of bonds in solids

and reduce to the set of smallest integers (hkl) Miller indices -

represent a set of parallel planesIf there are several equivalent nonparallel sets of planes: {hkl}Lecture 1 Andrei Sirenko, NJIT 37 Lecture 1 Andrei Sirenko, NJIT 38

Reciprocal latticeConstricting the reciprocal lattice from the direct lattice: Let a1, a2, a3 - primitive vectors of the direct lattice; T = n1a1 + n2a2 + n3a3Then reciprocal lattice can be generated using the primitive vectors

Some examples of reciprocal lattices 1. Reciprocal lattice to simple cubic lattice a1 = ax, a2 = ay, a3 = az V = a1(a2a3) = a3 b3 = (2/a)z

b1 = (2/a)x,

b2 = (2/a)y,

reciprocal lattice is also cubic with lattice constant 2/a 2. Reciprocal lattice to bcc latticewhere V = a1(a2a3) is the volume of the unit cell We have biaj = ij

a1 =

1 a ( x + y + z ) 2

a2 =

1 a(x y + z ) 2

1 1 a(x + y z ) V = a1 a 2 a 3 = a 3 2 2 2 2 (y + z ) b 2 = (x + z ) b 3 = 2 (x + y ) b1 = a a a a3 =39 Lecture 1 Andrei Sirenko, NJIT 40

Lecture 1


got

b1 =

2 (y + z ) a 2 (x + z ) a 2 (x + y ) a

Brillouin zones Determine all the perpendicular bisecting planes in the reciprocal lattice First Brillouin zone - the WignerSeitz cell of the reciprocal latticeSecond Brillouin zone: Higher Brillouin zones:

b2 = b3 =

but these are primitive vectors of fcc lattice So, the reciprocal lattice to bcc is fcc. Analogously, show that the reciprocal lattice to fcc is bcc

Lecture 1


41

Lecture 1


42

Brillouin zones of cubic lattices First BZ of bcc lattice First BZ of fcc lattice

SummaryReciprocal lattice is defined by primitive vectors:

A reciprocal lattice vector has the form G = hb1 + kb2 + lb3 It is normal to (hkl) planes of direct lattice First Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice Simple cubic cube; bcc Rhombic dodecahedron; fcc truncated octahedron (figures on the previous slide)

Lecture 1


43

Lecture 1


44

Solid State Physics Lecture 1-2 We already know: crystal lattice; 7 crystal systems; 14 Bravais lattices 32 point symmetry groups (crystal classes) Crystallographic directions and Miller indices; dhkl

Interatomic forces What holds a crystal together? Attractive electrostatic interaction between electrons and nuclei the force responsible for cohesion of solidsInteratomic potential V

Force:equilibrium position

F ( R) =

CONTINUE with: Inter-atomic forces and types of bonds in solids.

R0

V ( R ) R

binding energyAndrei R Interatomic distance Sirenko, NJIT

F(R) < 0 for R > R0 : attraction F(R) > 0 for R < R0 : repulsion46

Lecture 1


45

Lecture 1

Types of bonding I. Ionic crystals Usually involve atoms of strongly different electro-negativities (Example: alkali halides).

II. Covalent crystals Electron pair bond: usually one electron from each atom Electrons tend to be localized in part between the two atoms The spins of electrons in the bond are anti-parallel Gap between fully occupied and unoccupied states dielectrics or semiconductors Directionality of covalent bonds. Example: carbon Hybridization. 2s22p2 2s2px2py2pz : sp3 tetrahedral configuration Also possible sp2: 2s2px2py planar (graphite, fullerenes) remaining pz : interlayer -bonding Covalent polar bond (many compound semiconductors) intermediate case between non-polar and ionic bond. Described by effective ionic charge or fractional ionic character (between 0 and 1: 0 for Si or diamond, 0.94 for NaCl). Covalent bond is also strong, binding energies of several eV per atomLecture 1 Andrei Sirenko, NJIT 48

U ( R) = Nattractive (Coulomb)

e 2 A +N n 4 0 R Rrepulsive

Ionic bond is strong (binding energy - few eV/pair) hardness, high melting T electrons are strongly localized insulators in solid form Typical crystal structures: NaCl, CsClLecture 1


KCl: energy per molecule vs R (from Kittel)47

III. Metals Most elements in periodic table High electrical and thermal conductivity High density Considerable mechanical strength, but plasticity These properties can be explained considering the metallic type of bond Example: alkali metals single electron weakly bound to atom easily delocalized. In contrast to covalent bonding, electronic wave functions are very extended compared to interatomic distances. Why total energy is reduced ? Partially occupied electronic bands electrons can move freely Group II metals two s electrons should be fully occupied... but overlapped with empty p-states Transition metals: d-electrons are more localized form covalent-like bonds; s and p-electrons again form a common band Metals crystallize in closely packed structures (hcp, fcc, bcc)Lecture 1 Andrei Sirenko, NJIT 49 Lecture 1

Hexagonal and fcc close packed structure hcp fcc

both have coordination number 12 maximum possible for spheresAndrei Sirenko, NJIT 50

IV. Van der Waals bondsInert gases: outer electronic shells are full no ionic or covalent forces Weak interatomic forces due to quantum fluctuations of charge arising dipole moments cause a weak attractive force Can be described in the quantum-mechanical model of two linear oscillators (given in Kittel) results in R-6 dependence of potential Binding energy in order of 0.1 eV Crystal structures are fcc (electronic distribution is spherical, atoms pack as closely as possible) Van der Waals forces are also responsible for bonding in organic molecular crystals. Molecules are weakly bound; often form low-symmetry crystals They also exist in covalent or ionic crystals, but negligible

SummaryRepulsive interaction between atoms is primarily due to electrostatic repulsion of overlapping charge distributions and Pauli principle Several types of attractive forces: Ionic crystals electrostatic forces between "+" and "-" ions Covalent bond: overlap of charge distributions with antiparallel spin Metals: reduction of kinetic energy of electrons in free state compared to the localized state of a single atom Secondary forces (Van der Waals, hydrogen) become significant when the other bonds are impossible, e.g. in inert gases Physical properties are closely related to the type of bondingLecture 1 Andrei Sirenko, NJIT 52

V. Hydrogen bondsFormed between two strongly electronegative atoms (F, O, N) via H Example: iceLecture 1

Binding energy is also ~0.1 eVAndrei Sirenko, NJIT 51

Indexing system for crystal planes Condensed Matter Physics Crystals (continued)Andrei Sirenko Since crystal structures are obtained from diffraction experiments (in which particles diffract from planes of atoms), it is useful to develop a system for indexing lattice planes. We can use the lattice constants a1, a2, a3, but it turns out to be more useful to use what are called Miller Indices.IndexLecture 1 Andrei Sirenko, NJIT 53 Lecture 1 Andrei Sirenko, NJIT 54

Rules for determining Miller Indices (1) Find the intercepts on the axes in terms of the lattice constants a1, a2, a3. (2) Take the reciprocals of these numbers and then reduce to three integers having the same ratio, usually the smallest of the three integers. The result, listed as (hkl), is called the index of the plane.Lecture 1

Examples of Miller Indices

An example:

Intercepts: a, , Reciprocals: a/a, a/, a/ = 1, 0, 0 Miller index for this plane : (1 0 0) (note: this is the normal vector for this plane)55

Intercepts: a, a, Reciprocals: a/a, a/a, a/ = 1, 1, 0 Miller index for this plane : (1 1 0)Lecture 1

Intercepts: a,a,a Reciprocals: a/a, a/a, a/a = 1, 1, 1 Miller index for this plane : (1 1 1)56



Examples of Miller Indices Intercepts: 1/2a, a, Reciprocals: 2a/a, a/a, a/ = 2, 1, 0 Miller index for this plane : (2 1 0)

Notes on notation(hkl) might mean a single plane, or a set of planes If a plane cuts a negative axis, we have minus signs in the (hkl) (ie. (hkl)) Planes are denoted with curly brackets (hkl) A set of faces are denoted {hkl} The direction of a crystal (for example, along x for a cubic crystal) is denoted with [uvw] (ie. The [100] direction) In cubic crystals, the direction [hkl] is perpendicular to the plane (hkl) having the same indices, but this isnt necessarily true for other crystal systems

{001} face

[100] direction

[001] directionLecture 1 Andrei Sirenko, NJIT 57 Lecture 1 Andrei Sirenko, NJIT 58

Examples of common structures: (1) The Sodium Chloride (NaCl) Structure (LiH, MgO, MnO, AgBr, PbS, KCl, KBr) The NaCl structure is FCC The basis consists of one Na atom and one Cl atom, separated by one-half of the body diagonal of a unit cube There are four units of NaCl in each unit cube Atom positions: Cl : 000 ; 0; 0; 0 Na: ; 00; 00; 00 Each atom has 6 nearest neighbours of the opposite kind Often described as 2 interpenetrating FCC lattices

NaCl structureCrystal LiH MgO MnO NaCl AgBr PbS KCl KBrLecture 1

a 4.08 4.20 4.43 5.63 5.77 5.92 6.29 6.59

a

Lecture 1


59


60

(CsBr, CsI, RbCl, AlCo, AgZn, BeCu, MgCe, RuAl, SrTl) The CsCl structure is BCC The basis consists of one Cs atom and one Cl atom, with each atom at the center of a cube of atoms of the opposite kind There is on unit of CsCl in each unit cube Atom positions: Cs : 000 Cl : (or vice-versa) Each atom has 8 nearest neighbors of the opposite kindAndrei Sirenko, NJIT 61

(2) The Cesium Chloride (CsCl) structure

CsCl structureCrystal BeCu AlNi CuZn CuPd AgMg LiHg NH4Cl TlBr CsCl TlILecture 1

a 2.70 2.88 2.94 2.99 3.28 3.29 3.87 3.97 4.11 4.20Why are the a values smaller for the CsCl structures than for the NaCl (in general)?Andrei Sirenko, NJIT 62

a

Lecture 1

Closed-packed structures Closed-packed structures There are an infinite number of ways to organize spheres to maximize the packing fraction.The centers of spheres at A, B, and C positions (from Kittel) There are different ways you can pack spheres together. This shows two ways, one by putting the spheres in an ABAB arrangement, the other with ACAC. (or any combination of the two works)63 Lecture 1 Andrei Sirenko, NJIT 64

(or, what does stacking fruit have to do with solid state physics?)Lecture 1 Andrei Sirenko, NJIT

Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl

(3) The Hexagonal Closed-packed (HCP) structureThe HCP structure is made up of stacking spheres in a ABABAB configuration The HCP structure has the primitive cell of the hexagonal lattice, with a basis of two identical atoms Atom positions: 000, 2/3 1/3 (remember, the unit axes are not all perpendicular) The number of nearest-neighbours is 12. Rotated three times

The FCC and hexagonal closed-packed structures (HCP) are formed from packing in different ways. FCC (sometimes called the cubic closed-packed structure, or CCP) has the stacking arrangement of ABCABCABC HCP has the arrangement ABABAB. [1 1 1][0 0 1] HCP FCC (CCP) (looking along [111] direction

ABAB sequence

ABCABC sequence

Lecture 1


65

Lecture 1


66

Conventional HCP unit cell

HCP and FCC structuresCrystal He The hexagonal-closed packed (HCP) and FCC structures both have the ideal packing fraction of 0.74 (Kepler figured this out hundreds of years ago) The ideal ratio of c/a for this packing is (8/3)1/2 = 1.633 (see Problem 3 in Kittel) Why arent these values perfect for real materials? Why would real materials pick HCP or FCC (what does it matter if they pack the same?)Andrei Sirenko, NJIT

c/a 1.633 1.581 1.623 1.586 1.861 1.886 1.622 1.570 1.594 1.592 1.58667

Be Mg Ti Zn Cd Co Y Zr Gd Lu

Lecture 1

Documents

Lecture1-SSP-2007